# Aircraft Geometry and Meshing with Common Language Schema CPACS for Variable-Fidelity MDO Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- L0:
- handbook methods, based on statistics and/or empirical design rules;
- L1:
- based on simplified physics, can model and capture a limited amount of effects. For example, the linearized-equation models, the Vortex-Lattice Method (VLM) or the panel method in aerodynamics;
- L2:
- based on accurate physics representations. For example, the non-linear analysis, Euler-based CFD;
- L3:
- represents the highest end simulations, usually used to capture detailed local effects, but do not allow wide exploration of the design space due to computational cost. Additionally, the modeling may require extensive ad hoc manual intervention. For example, the highest fidelity methods, RANS-based CFD.

#### 1.1. Background

`cpacs.toolspecific`) to handle parameters for the computational models, which are relevant only for a specific tool [20].

#### 1.2. Aerodynamic Model Description

## 2. CPACS File Description

#### 2.1. The CPACS Hierarchical Data Definition Structure

`profileUID`s can be stored in the CPACS geometry definition and from these, a list of wing

`elements`. An

`element`is defined by its

`profileUID`and a transformation: the scaling along coordinate directions, a 3D rotation and a translation. Two such elements define a

`section`, the

`positioning`of which is effected by a

`length`,

`sweep angle`and a

`dihedral angle`. The sections are assembled to form a wing, to which it is possible once more to apply a transformation. Symmetries can be used to create instances of wings already defined, A single CPACS file holds a set of named lifting surfaces defined in this way. It must be noted that a given wing geometry allows multiple distinct CPACS definitions.

#### 2.2. The CPACS Control Surface Definition

`componentSegment`first needs to be created. Each

`componentSegment`is defined from two, not necessarily contiguous, wing elements. Each wing must have at least one

`componentSegment`to define the wing structure, fuel tanks, control surfaces, etc. Each corner of the outer shape of the control surface is defined by its relative position in the span- and chord-wise directions of the

`componentSegment`, as shown in Figure 5, requiring eight values to be specified. For TEDs, corner points that are not explicitly defined lie on the trailing edge of the wing. In addition to these points, the

`hingeLine`must also be defined, by the relative position of its inner and outer points in the span- and chord-wise directions, as well as their “vertical” position from 0 = lower wing surface to 1 = upper wing surface.

`step`, is defined by an angle of rotation around the hinge line and a translation of the rotated surface, to allow the definition of flap movements. A deployment is then defined by interpolation in the table of

`steps`.

`outerShape`node affiliated with each defined control surface. The information is then interpreted by the different models in the CFD tools.

## 3. Geometry and CPACS Interfaces for Variable Fidelity Tools

`Sumo`/

`TetGen`[24,25], will be discussed next. Tornado is a VLM implementation for assessing aero-forces and moments on rigid lifting surfaces.

`Sumo`/

`TetGen`is an automatic volume mesh generator for CFD. It is fully automatic for the generation of isotropic tetrahedral grids for Euler solvers. Its

`Pentagrow`[8] module provides semi-automatic mesh generation for RANS.

#### 3.1. CPACS-Tornado Interface

- Aircraft configuration visualization including fuselage representation and control surface identifications;
- Fast MEX-compiled version of core-functions for matrix computations;
- All-moving surfaces and overlapped movable surfaces.

#### PyTornado: A VLM Solver with Native CPACS Compatibility

- A Python wrapper, dedicated to high-level tasks such as communication with CPACS, pre- and post-processing for VLM, as well as visualization of the model and generated results; see Figure 7a,b,
- The actual VLM solver, re-structured and re-written in C++ from the MATLAB Tornado VLM solver with performance in mind.

#### 3.2. CPACS-Sumo Interface

`Sumo`for L2 analysis and its interface with CPACS.

`Sumo`can also generate RANS (L3) meshes with

`Pentagrow`[8]; note that the RANS simulations are only used for validation in this paper (see Section 5.1).

#### 3.2.1. `Sumo`: A Gateway from CPACS to Higher-Fidelity Aerodynamics

`Sumo`[24] is a graphic tool for rapid modeling of aircraft geometries and automatic unstructured surface mesh generation. It is not a full-fledged CAD system, but rather an easy-to-use sketchpad, highly specialized towards aircraft configurations in order to streamline the workflow. Isotropic tetrahedral volume meshes for Euler computation can be generated from the surface mesh, by the tetrahedral mesh generator

`TetGen`[25].

`Pentagrow`[8] module in

`Sumo`after the surface mesh is generated, before creation of the volume mesh by

`TetGen`.

`Pentagrow`sets up the prismatic element layers on the configuration surface from a configuration file with a list of user-defined parameters such as the first cell height, the total number of layers, the growth rate, etc. The volume mesh can be exported in various formats including CGNS (the CFD General Notation System),

`TetGen`’s plain ASCII format and native formats for the CFD solvers Edge [28] and SU2 [29,30,31]. Mesh examples are shown in Figure 9a,b.

#### 3.2.2. The Interface CPACS2SUMO

`Sumo`[24] native

`.smx`file by the CPACS2SUMO Python converter without manual intervention. This conversion is relatively straightforward since both formats define aircraft in a similar way. Fuselage and wings are created from a gathering of sections placed in a certain order. Each section is defined by a 2D profile written as a list of points. Then, these profiles can be scaled, rotated and translated to form the desired shape. Figure 10 shows how

`Sumo`represents a wing as a stack of airfoils. The 3D wing surface is lofted from the sections by Bézier or B-spline surfaces.

`Sumo`uses the order of the sections as they appear in the file.

`Sumo`(Figure 11), transformation by scaling, rotation and translation of an entity is executed at one level and with limitation. For example, a

`Sumo`fuselage profile is assumed perpendicular to the x-axis. Furthermore, CPACS and

`Sumo`formats use different definitions of 3D rotation angles.

## 4. Flow Solvers

## 5. Applications

#### 5.1. Aerodynamic Results Comparison

`Pentagrow`using the same

`Sumo`surface mesh for the L2 Euler computations, which has 792,900 triangles on the surface. The RANS mesh has 8.2 million cells. The first layer height is $3.8\times {10}^{-6}$ (${y}^{+}=1$); the growth rate is 1.2; the number of layers is 40, with the corresponding Reynolds number 32.4 million of the reference chord 3.7317 m. The airspeed is 192 m/s, which corresponds to Mach = 0.6 and altitude = 5000 m. Figure 14 shows the computed ${y}^{+}$ diagram over the reference aircraft at $\alpha ={1}^{\circ}$ with airspeed 192 m/s and Reynolds number 32.4 million.

#### 5.2. Multi-Fidelity Aerodynamics for Data Fusion

#### 5.3. Aero-Data for Low Speed by the L1 Tool

#### 5.3.1. Sizing the Fin and Rudder for the One-Engine-Out Case

#### 5.3.2. Handling Qualities

## 6. Euler Computation for Various Control Surface Models

#### 6.1. Modeling Movable Surfaces

`SU2_DEF`built-in mesh deformation function is then used to deform the mesh around the elevator locations on the horizontal tail (see Figure 19a). An FFD box is defined at the elevator locations. The FFD box of control points can be rotated around the hinge line. Owing to the affine invariance of the map from control points to the mesh, the surface mesh follows. According to the authors’ experience, with a deflection of less than eight degrees, the deformed mesh is still well formed enough to function for Euler simulation isotropic grids.

`Sumo`are described in [40].

`Sumo`surface mesh for a deflection angle of six degrees. Both mesh deformation by FFD (Figure 19a) and morphing of the camber lines produce gapless meshes according to the control surface deflections. The morphing technology in addition supports a user-defined transition zone (length and type) to obtain a smoother surface and avoid bad tetrahedral cells. The smooth transition feature makes the trailing edge morphing technology possible for a RANS simulation; however, the mesh deformation by FFD tends to produce high aspect ratio cells at the deformed junctions, so the mesh may work well only for coarser Euler simulations.

#### 6.2. Results Comparison

- Mesh-Def(orm): Mesh deformation using FFD;
- Morph. (cs): Morphing the control surfaces by
`Sumo`; - Transp. b.c.: transpiration boundary conditions in Edge.

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGILE | Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts |

API | Application Programming Interface |

CAD | Computer Aided Design |

CFD | Computational Fluid Dynamics |

CPACS | The Common Parametric Aircraft Configuration Schema |

CST | Class-Shape function Transformation |

DES | Detached Eddy Simulation |

FFD | Free-Form Deformation |

GUI | Graphic User Interface |

LES | Large-Eddy Simulation |

MAC | Mean Aerodynamic Chord |

MDA | Multidisciplinary Analysis |

MDO | Multidisciplinary Design and Optimization |

RANS | Reynolds-Averaged Navier–Stokes equations |

TAS | True Air Speed |

TE(D) | Trailing Edge (Device) |

UI | User Interface |

VLM | Vortex Lattice Method |

XML | Extensible Markup Language |

Symbols | |

$\alpha $ or AoA | Angle of Attack (deg) |

${\delta}_{e}$ | Elevator deflection angle (deg) |

$\theta $ | Attitude angle (deg) |

q | Pitch rate (deg/s) |

${C}_{p}$ | Pressure Coefficient (-) |

${C}_{L}$ | Lift coefficient (-) |

${C}_{D}$ | Drag coefficient (-) |

${C}_{m}$ | Pitching moment coefficient (-) |

## References

- Voskuijl, M.; de Klerk, J.; van Ginneken, D. Flight Mechanics Modelling of the Prandtl Plane for Conceptual and Preliminary Design; Springer: London, UK, 2012; Volume 66. [Google Scholar]
- Voskuijl, M.; La Rocca, G.; Dircken, F. Controllability of blended wing body aircraft. In Proceedings of the 26th Congress of International Council of the Aeronautical Sciences, Anchorage, AK, USA, 14–19 September 2008. [Google Scholar]
- Fengnian, T.; Voskuijl, M. Automated Generation of Multiphysics Simulation Models to Support Multidisciplinary Design Optimization. Adv. Eng. Inform.
**2015**, 29, 1110–1125. [Google Scholar] - Foeken, M.J.; Voskuijl, M. Knowledge-Based Simulation Model Generation for Control Law Design Applied to a Quadrotor UAV. Math. Comput. Model. Dyn. Syst.
**2010**, 16, 241–256. [Google Scholar] [CrossRef] - Raymer, D. Conceptual design modeling the RDS-professional aircraft design software. In Proceedings of the AIAA Aerospace Sciences Meeting, Orlando, FL, USA, 4–7 January 2011. [Google Scholar]
- Anemaat, W.A.; Kaushik, B. Geometry design assistant for airplane preliminary design. In Proceedings of the 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, FL, USA, 4–7 January 2011. [Google Scholar]
- Gloudemans, J.R.; Davis, P.C.; Gelausen, P.A. A rapid geometry modeler for conceptual aircraft. In Proceedings of the 34th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 15–18 January 1996. [Google Scholar]
- Tomac, M. Towards Automated CFD for Engineering Methods in Aircraft Design. Ph.D. Thesis, Royal Institute of Technology, KTH, Stockholm, Sweden, 2014. [Google Scholar]
- CPACS—A Common Language for Aircraft Design. Available online: http://www.cpacs.de/ (accessed on 7 February 2018).
- Böhnke, D.; Nagel, B.; Zhang, M.; Rizzi, A. Towards a collaborative and integrated set of open tools for aircraft design. In Proceedings of the 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Grapevine, TX, USA, 7–10 January 2013; pp. 7–10. [Google Scholar]
- SUAVE—An Aerospace Vehicle Environment for Designing Future Aircraft. Available online: http://suave.stanford.edu/ (accessed on 20 March 2018).
- Lukaczyk, T.; Wendorff, A.; Botero, E.; MacDonald, T.; Momose, T.; Variyar, A.; Vegh, J.M.; Colonno, M.; Economon, T.; Alonso, J.J.; et al. SUAVE: An Open-Source Environment for Multi-Fidelity Conceptual Vehicle Design. In Proceedings of the 16th AIAA Multidisciplinary Analysis and Optimization Conference, Dallas, TX, USA, 22–26 June 2015. [Google Scholar]
- MacDonald, T.; Clarke, M.; Botero, E.M.; Vegh, J.M.; Alonso, J.J. SUAVE: An Open-Source Environment Enabling Multi-fidelity Vehicle Optimization. In Proceedings of the 18th AIAA Multidisciplinary Analysis and Optimization Conference, Denver, CO, USA, 5–9 June 2017. [Google Scholar]
- Zhang, M.; Bartoli, N.; Jungo, A.; Lammen, W.; Baalbergen, E. Data Fusion and Aerodynamic Surrogate Modeling for Handling Qualities Analysis. Prog. Aerosp. Sci. Spec. Issue
**2018**. submitted. [Google Scholar] - AGILE EU Project Portal. Available online: http://www.agile-project.eu (accessed on 15 January 2018).
- Seider, D.; Fischer, P.; Litz, M.; Schreiber, A.; Gerndt, A. Open Source Software Framework for Applications in Aeronautics and Space. In Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA, 3–10 March 2012. [Google Scholar]
- Baalbergen, E.; Kos, J.; Louriou, C.; Campguilhem, C.; Barron, J. Streamlining cross-organisation product design in aeronautics. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2016**. [Google Scholar] [CrossRef] - Moerland, E.; Ciampa, P.D.; Zur, S.; Baalbergen, E.; D’Ippolito, R.; Lombardi, R. Collaborative Architecture supporting the next generation of MDO within the AGILE Paradigm. Prog. Aerosp. Sci. Spec. Issue
**2018**. submitted. [Google Scholar] - Van Gent, I.; Aigner, B.; Beijer, B.; Jepsen, J.; Rocca, G.L. Knowledge architecture supporting the next generation of MDO in the AGILE paradigm. Prog. Aerosp. Sci. Spec. Issue
**2018**. submitted. [Google Scholar] - Jungo, A.; Vos, J.; Zhang, M.; Rizzi, A. Benchmarkig New CEASIOM with CPACS Adoption for Aerodynamic Analysis and Flight Simulation. Aircr. Eng. Aerosp. Technol.
**2017**, 90. [Google Scholar] [CrossRef] - TIGL—Geometry Library to Process Aircraft Geometries in Pre-Design. Available online: https://github.com/DLR-SC/tigl (accessed on 30 November 2017).
- CPACS Documentation. Available online: https://github.com/DLR-LY/CPACS/tree/develop/documentation (accessed on 20 April 2018).
- Melin, T. Using Internet Interactions in Developing Vortex Lattice Software for Conceptual Design. Ph.D. Thesis, Department of Aeronautics, Royal Institute of Technology, KTH, Stockholm, Sweden, 2003. [Google Scholar]
- Tomac, M.; Eller, D. From Geometry to CFD Grids: An Automated Approach for Conceptual Design. Prog. Aerosp. Sci.
**2011**, 47, 589–596. [Google Scholar] [CrossRef] - Si, H. TetGen: A Quality Tetrahedral Mesh Generator and 3D Delaunay Triangulator; Technical Report, User’s Manual; Technical Report No. 13; Numerical Mathematics and Scientific Computing, Weierstrass Institute for Applied Analysis and Stochastics (WIAS): Berlin, Germany, 2013. [Google Scholar]
- Anderson, J.D. Modern Compressible Flow with Historical Perspective, 3rd ed.; McGraw-Hill: New York, NY, USA, 2004. [Google Scholar]
- Katz, J.; Plotkin, A. Low-Speed Aerodynamics: From Wing Theory to Panel Methods; McGraw-Hill, Inc.: New York, NY, USA, 1991. [Google Scholar]
- Eliasson, P. Edge, a Navier-Stokes Solver for Unstructured Grids. In Finite Volumes for Complex Applications III; Elsevier: Amsterdam, The Netherlands, 2002; pp. 527–534. [Google Scholar]
- Palacios, F.; Colonno, M.R.; Aranake, A.C.; Campos, A.; Copeland, S.R.; Economon, T.D.; Lonkar, A.K.; Lukaczyk, T.W.; Taylor, T.W.R.; Alonso, J. Stanford University Unstructured (SU2): An open-source integrated computational environment for multi-Physics simulation and design. In Proceedings of the 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Grapevine, TX, USA, 7–10 January 2013. AIAA 2013-0287. [Google Scholar]
- Economon, T.D.; Palacios, F.; Copeland, S.R.; Lukaczyk, T.W.; Alonso, J.J. SU2: An Open-Source Suite for Multiphysics Simulation and Design. AIAA J.
**2015**, 54, 828–846. [Google Scholar] [CrossRef] - Palacios, F.; Economon, T.D.; Wendorff, A.D.; Alonso, J. Large-Scale Aircraft Design Using SU2. In Proceedings of the 53rd AIAA Aerospace Sciences Meeting, AIAA 2015-1946, Kissimmee, FL, USA, 5–9 January 2015. [Google Scholar]
- Griva, I.; Nash, S.G.; Sofer, A. Linear and Nonlinear Optimization, 2nd ed.; Society for Industrial Applied Mathematics: Philadelphia, PA, USA, 2009. [Google Scholar]
- Yousefi, K.; Razeghi, A. Determination of the Critical Reynolds Number for Flow over Symmetric NACA Airfoils. In Proceedings of the 2018 AIAA Aerospace Sciences Meeting, AIAA 2018–0818, Kissimmee, FL, USA, 8–12 January 2018. [Google Scholar]
- Forrester, A.; Keane, A. Engineering Design via Surrogate Modelling: A Practical Guide; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Federal Aircraft Administration. CFR 25.149; Technical Report, Retrieved 18 January 2018; Federal Aircraft Administration: Washington, DC, USA, 2018.
- Torenbeek, E. Synthesis of Subsonic Airplane Design; Delft University Press: Delft, The Netherlands, 1982. [Google Scholar]
- Etkin, B.; Reid, L.D. Dynamics of Flight: Stability and Control; John Wiley & Sons: Hoboken, NJ, USA, 1996. [Google Scholar]
- Pfeiffer, T.; Nagel, B.; Böhnke, D.; Voskuijl, M.; Rizzi, A. Implementation of a Heterogeneous, Variable-Fidelity Framework for Flight Mechanics Analysis in Preliminary Aircraft Design; Deutscher Luft-und Raumfahrtkongress: Bremen, Germany, 2011. [Google Scholar]
- Sederberg, T.W.; Parry, S.R. Free-Form Deformation of Solid Geometric Models. In Proceedings of the SIGGRAPH ’86 13th Annual Conference on Computer Graphics and Interactive Techniques, Atlanta, GA, USA, 1–5 August 1986; Volume 20, pp. 151–160. [Google Scholar]
- Zhang, M. Contributions to Variable Fidelity MDO Framework for Collaborative and Integrated Aircraft Design. Ph.D. Thesis, Royal Institute of Technology KTH, Stockholm, Sweden, 2015. [Google Scholar]
- Kulfan, B. Universal Parametric Geometry Representation Method. J. Aircr.
**2008**, 45, 142–158. [Google Scholar] [CrossRef]

**Figure 1.**The reference aircraft, rendered by the CPACS visualization tool TIGLViewer [21].

**Figure 2.**The airfoil details of the reference aircraft. (

**a**) The airfoils of the reference aircraft; (

**b**) the thickness and camber information of the airfoils.

**Figure 3.**The CPACS hierarchical structure (image from the CPACS website [9]).

**Figure 4.**Adapted from CPACS documentation [9]; schema for the construction of an aircraft wing from its XML file definition.

**Figure 6.**Tornado partition layout with control surfaces for the reference aircraft, imported from CPACS.

**Figure 7.**Panel distributions and the ${C}_{p}$ visualization in PyTornado for a box-wing aircraft, imported from CPACS. (

**a**) Panel layout; (

**b**) the ${C}_{p}$ simulation for U = 100 m/s, $\alpha ={5}^{\circ}$.

**Figure 9.**The surface and volume meshes of the reference aircraft generated by

`Sumo`with

`TetGen`and

`Pentagrow`. (

**a**)

`Sumo`surface mesh; (

**b**)

`Sumo-Pentagrow`RANS mesh.

**Figure 12.**Euler simulations for the reference aircraft for ${C}_{L}$, ${C}_{D}$ and ${C}_{m}$ computed by Edge and SU2 for meshes “mesh-2p” and “mesh-4p”, at a flight altitude of 10,000 m, Mach 0.78 and Mach 0.9 respectively. (

**a**) Lift coefficient ${C}_{L}$ for Mach number of 0.78 and 0.9; (

**b**) drag coefficient ${C}_{D}$ for Mach numbers of 0.78 and 0.9; (

**c**) pitching moment coefficient ${C}_{m}$ for Mach numbers of 0.78 and 0.9.

**Figure 13.**L1, L2 and L3 simulations for the test case aircraft, $\alpha $ sweep at Mach = 0.6, altitude = 5000 m and $\beta ={0}^{\circ}$.

**Figure 14.**The ${y}^{+}$ diagram over the reference aircraft at $\alpha ={1}^{\circ}$ with airspeed 192 m/s and Reynolds number 32.4 million.

**Figure 15.**Surrogate model results of the reference aircraft for ${C}_{L}$, ${C}_{D}$ and ${C}_{m}$, with no elevator deflection. Notations: dot: Tornado (L1) samples; cross: SU2 (L2) Euler samples; line: the response surfaces. (

**a**,

**c**,

**e**) The response surfaces and sampled data over the flight envelope. (

**b**,

**d**,

**f**) The cuts for Mach number 0.5 (black) and 0.78 (blue) from the response surfaces and their corresponding sampled data; the results are also shown in [14]. (

**a**) Lift coefficient ${C}_{L}$ surface and the sampled data; (

**b**) fused lift coefficient ${C}_{L}$ for Mach numbers of 0.5 and 0.78; (

**c**) drag coefficient ${C}_{D}$ surface and the sampled data; (

**d**) fused drag coefficient ${C}_{D}$ for Mach numbers of 0.5 and 0.78; (

**e**) pitching moment coefficient ${C}_{m}$ surface and the sampled data; (

**f**) fused pitching moment coefficient ${C}_{m}$ for Mach 0.5 and 0.78.

**Figure 18.**Flight time domain simulation from PHALANX, trimmed flight with True Air Speed (TAS) = 130 m/s at sea level.

**Figure 19.**The mesh generated by

`Sumo`with different modeling technologies to be computed for elevator deflection at 6${}^{\circ}$. (

**a**) Surface mesh with mesh deformed by FFD on the elevator for ${\delta}_{e}=6$ degrees; (

**b**) surface mesh with morphed elevator modeled by

`Sumo`for ${\delta}_{e}=6$ degrees, which includes a linear type 5% transition zone of the morphing elevator length; (

**c**) tailplane and the elevators marked in the surface mesh in

`Sumo`.

**Figure 20.**The demonstration of the automatically gapless movable surfaces morphing technology by

`Sumo`. (

**a**) The geometric parameters describing the morphing airfoil [40], morphing leading edge and trailing edge and a fixed central area using a variable camber; (

**b**) illustration of the morphing trailing edge with the morphing zone and transition zones modeled in

`Sumo`.

**Figure 21.**Euler solutions for elevator deflection ${\delta}_{e}={6}^{\circ}$ at Mach 0.78, flight altitude 10,000 m, $\alpha ={0}^{\circ}$, by three control surface modeling methods, for SU2 and Edge.

**Figure 22.**${C}_{p}$ contours of Euler solutions for the elevator deflection ${\delta}_{e}={6}^{\circ}$ at Mach 0.78, flight altitude 10,000 m, $\alpha ={0}^{\circ}$, by three control surface modeling methods, SU2 and Edge.

**Table 1.**Result for different modeling of elevator deflections using different solvers, at Mach 0.78, flight altitude 10,000 m, $\alpha ={0}^{\circ}$.

Model Type | Solver | ${\mathit{C}}_{\mathit{L},\mathit{\delta}}$ (deg) | ${\mathit{C}}_{\mathit{m},\mathit{\delta}}$ (deg) |
---|---|---|---|

Mesh-deform | SU2 | 0.0092 | −0.0399 |

Morph. cs | SU2 | 0.0130 | −0.0565 |

Morph. cs | Edge | 0.0117 | −0.0557 |

Transp. b.c. | Edge | 0.0095 | −0.0411 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Jungo, A.; Gastaldi, A.A.; Melin, T. Aircraft Geometry and Meshing with Common Language Schema CPACS for Variable-Fidelity MDO Applications. *Aerospace* **2018**, *5*, 47.
https://doi.org/10.3390/aerospace5020047

**AMA Style**

Zhang M, Jungo A, Gastaldi AA, Melin T. Aircraft Geometry and Meshing with Common Language Schema CPACS for Variable-Fidelity MDO Applications. *Aerospace*. 2018; 5(2):47.
https://doi.org/10.3390/aerospace5020047

**Chicago/Turabian Style**

Zhang, Mengmeng, Aidan Jungo, Alessandro Augusto Gastaldi, and Tomas Melin. 2018. "Aircraft Geometry and Meshing with Common Language Schema CPACS for Variable-Fidelity MDO Applications" *Aerospace* 5, no. 2: 47.
https://doi.org/10.3390/aerospace5020047